**SymMonCat** is the cartesian 2-category of symmetric monoidal categories, braided monoidal functors, and monoidal natural transformations. The terminal symmetric monoidal category **1** has one object $I$ and $I \otimes I = I$.

A category enriched over a monoidal category $V$ assigns to each pair of objects $X, Y$ an object hom$(X,Y)$ in $V$ and to each object $X$ a morphism $id_X:I \to \mbox{hom}(X,X)$ in $V$.

When $V = $ **SymMonCat**, the morphism $id_X:1 \to \mbox{hom}(X,X)$ is a braided monoidal functor; since monoidal functors preserve the monoidal unit and tensor product, it must map the unit $I$ in **1** to the unit $I$ in hom$(X,X)$.

Is there a different way of enriching over **SymMonCat** such that $id_X$ does not pick out the monoidal unit (other than considering it a subcategory of **Cat**)?

coproductof symmetric monoidal categories. In particular, you should not expect the kind of "hom-tensor adjunction" that you usually want when writing down a good theory of enriched categories. (continued) $\endgroup$ – Theo Johnson-Freyd Nov 9 '10 at 22:201more comments